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Matrix Addition

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Matrix Addition as the name suggests in the article, explores the addition of matrices and it is one of the fundamental operations in the field of Linear Algebra. Matrix Addition is very useful in various fields such as data analysis, computer graphics, image processing, cryptography, operations research, machine learning, artificial intelligence, etc. In this article, we will explore the concept of Matrix Addition, including its properties and solved examples as well.

Matrix Definition

A matrix is a rectangular array or set of elements. The Matrix can be defined as an m×n element in the form of m horizontal lines (rows), and n vertical lines (columns) known as the m*n order matrix. Elements can be real, complex, or unknown numbers. An m×n matrix is given as follows:

\bold{\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}}

In the above figure, a matrix of order m×n is shown where i and j represent the element’s exact position (i,j) i.e., ith row and jth column.

What is Matrix Addition?

Matrix addition is the operation defined on the matrix to add two matrices to get a single matrix. Let’s suppose two matrices A and B, such A = [aij] and B = [bij], then their addition A + B is defined as [aij + bij], where ij represents the element in ith row and jth column.

 

 Let’s consider the following examples for better understanding.

Example: For matrix \bold{A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} }        and \bold{B = \begin{bmatrix} 7 & 8 & 9 \\ 10 & 11 & 12 \end{bmatrix} }       , calculate A + B.

Solution:

 A + B = \begin{bmatrix} 1 + 7 & 2 + 8 & 3 + 9 \\ 4 + 10 & 5 + 11 & 6 + 12 \end{bmatrix}

\Rightarrow A + B = \begin{bmatrix} 8 & 10 & 12 \\ 14 & 16 & 18 \end{bmatrix}

Properties of Matrix Addition

There are various unique properties of matrix addition. We will be discussing the below-mentioned properties:

  • Closure Property
  • Commutative Property
  • Associative Property
  • Additive Identity Property
  • Additive Inverse Property

Let’s consider three matrices A, B, and C of the same order m×n, as to add two Matrices they need to have the same order, simply add the corresponding element of each Matrix. Let’s discuss the addition property of Matrix in detail.

Closure Property of Matrix Addition

A matrix can be added with another matrix if and only if the order of matrices is the same. The addition will take place between the elements of the matrices. The resultant matrix will also be of the same order. That is [A]m×n + [B]m×n = [C]m×n 

Example:

\bold{A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2\times3}} \\ \bold{\text{and } B = \begin{bmatrix} 7 & 8 & 9 \\ 10 & 11 & 12 \end{bmatrix}_{2\times3}} \\ \bold{\Rightarrow C = A + B = \begin{bmatrix} 8 & 10 & 12 \\ 14 & 16 & 18 \end{bmatrix}_{2\times3}}

Commutative Property of Matrix Addition

Commutative Property states that any two matrices of the same order can be added in any way i.e., the result of the sum of two matrices doesn’t depend on the order of the matrix in matrix addition. Suppose there are two matrices A and B of the same order m*n, then the commutative property of matrix addition states that: A + B = B + A

Example : For matrix \bold{A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \text{and } B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}}

A + B =\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \\ \Rightarrow A + B = \begin{bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

B + A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\\ \Rightarrow B + A = \begin{bmatrix} (5 + 1) & (6 + 2) \\ (7 + 3) & (8 + 4) \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

Thus, A + B = B + A., which demonstrates the commutative property of matrix addition.

Associative Property of Matrix Addition

Similarly, If three matrices have the same order then their position does not matter in addition. Suppose there are three matrices A, B, and C of order m*n, then the associative property of matrix addition states that: A + (B + C) = (A + B) + C

Example: For  \bold{A =\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}, B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix}, \text{and } C =  \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \\ \end{bmatrix}}

(A + B) + C = \begin{bmatrix}    (a_{11} + b_{11}) & (a_{12} + b_{12}) \\    (a_{21} + b_{21}) & (a_{22} + b_{22}) \\ \end{bmatrix} + \begin{bmatrix}    c_{11} & c_{12} \\    c_{21} & c_{22} \\ \end{bmatrix} \\ \Rightarrow (A + B) + C= \begin{bmatrix}    [(a_{11} + b_{11}) + c_{11}] & [(a_{12} + b_{12}) + c_{12}] \\    [(a_{21} + b_{21}) + c_{21}] & [(a_{22} + b_{22}) + c_{22}] \\ \end{bmatrix}

A + (B + C)= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} + \begin{bmatrix} (b_{11} + c_{11}) & (b_{12} + c_{12}) \\ (b_{21} + c_{21}) & (b_{22} + c_{22}) \\ \end{bmatrix}\\ \Rightarrow A + (B + C) = \begin{bmatrix} [(a_{11} + b_{11}) + c_{11}] & [(a_{12} + b_{12}) + c_{12}] \\ [(a_{21} + b_{21}) + c_{21}] & [(a_{22} + b_{22}) + c_{22}] \\ \end{bmatrix}

Therefore, (A + B) + C = A + (B + C), which demonstrates the associative property of matrix addition.

Additive Identity Property of Matrix Addition

We have discussed zero Matrix that O matrix can be added to any matrix for the same result. According to the additive identity property of matrix addition, for a given matrix A of order m*n, there exists an m×n matrix O such that: A + O = A = O + A

Here, O is the m×n order zero Matrix.

Example: Let A be a 2×2 matrix, and let I be the 2×2 identity matrix. We want to show that A + O = A = O + A.

Solution:

A = \begin{bmatrix}   a_{11} & a_{12} \\   a_{21} & a_{22} \\ \end{bmatrix} \text{and } O = \begin{bmatrix}   0 & 0 \\   0 & 0 \\ \end{bmatrix}

A + O = \begin{bmatrix}   a_{11} & a_{12} \\   a_{21} & a_{22} \\ \end{bmatrix} + \begin{bmatrix}   0 & 0 \\   0 & 0 \\ \end{bmatrix} \\ \Rightarrow A + O = \begin{bmatrix}   (a_{11} + 0) & (a_{12} + 0) \\   (a_{21} + 0) & (a_{22} + 0) \\ \end{bmatrix} \\ \Rightarrow A + O = \begin{bmatrix}   a_{11}  & a_{12} \\   a_{21} & a_{22}  \\ \end{bmatrix}

O + A = \begin{bmatrix}   0 & 0 \\   0 & 0 \\ \end{bmatrix} + \begin{bmatrix}   a_{11} & a_{12} \\   a_{21} & a_{22} \\ \end{bmatrix}\\ \Rightarrow O + A = \begin{bmatrix}   (0 + a_{11}) & (0 + a_{12}) \\   (0 + a_{21}) & (0 + a_{22}) \\ \end{bmatrix} \\ \Rightarrow O + A = \begin{bmatrix}   a_{11}  & a_{12} \\   a_{21} & a_{22}  \\ \end{bmatrix}

So, if you add a matrix to a zero matrix, then you get the original Matrix.

Additive Inverse Property of Matrix Addition

There is a rule in Matrix that the inverse of any matrix A is –A of the same order. In simple words, for a given matrix A of order m*n, there exists a unique matrix B such that: A + B = O

Note: This matrix B is equal to –A i.e. B = -A

Therefore, A + (-A) = O

Example: Let A be a 2×2 matrix, and let -A be the additive inverse of A. We want to show that A + (-A) = O, where O is the 2×2 zero matrix.

Solution:

A = \begin{bmatrix}   a_{11} & a_{12} \\   a_{21} & a_{22} \\ \end{bmatrix}

The additive inverse of A, denoted -A, is given by:

-A = \begin{bmatrix}   -a_{11} & -a_{12} \\   -a_{21} & -a_{22} \\ \end{bmatrix}

Now, let’s compute A + (-A).

A + (-A) = \begin{bmatrix}   a_{11} & a_{12} \\   a_{21} & a_{22} \\ \end{bmatrix} + \begin{bmatrix}   -a_{11} & -a_{12} \\   -a_{21} & -a_{22} \\ \end{bmatrix} \\ \Rightarrow A + (-A) = \begin{bmatrix}   (a_{11} + (-a_{11})) & (a_{12} + (-a_{12})) \\   (a_{21} + (-a_{21})) & (a_{22} + (-a_{22})) \\ \end{bmatrix} \\ \Rightarrow A + (-A) = \begin{bmatrix}    0 & 0 \\    0 & 0 \\ \end{bmatrix} = O

Therefore, A + (-A) = O, which demonstrates the additive inverse property of matrix addition.

What is Matrix Subtraction?

As we add two or more matrices in the same way we can subtract two matrices, if they are square matrices of the same order. Matrix addition is similar to matrix subtraction, we can assume that matrix subtraction is the addition of one matrix with the additive inverse of the second matrix.

If we have two matrices A and B the subtraction of A and B can be understood as, the addition of A and (-B), i.e.

A – B = A + (-B)

For further understanding study the following example,

Example: Let A and B be 2×2 matrices, where A = \begin{bmatrix}   2 & 4 \\   6 & 8 \\ \end{bmatrix}   and B = \begin{bmatrix}   1 & 3 \\   5 & 7 \\ \end{bmatrix}   then find A – B.

Solution:

Given,

A = \begin{bmatrix}   2 & 4 \\   6 & 8 \\ \end{bmatrix}

B = \begin{bmatrix}   1 & 3 \\   5 & 7 \\ \end{bmatrix}

-B = \begin{bmatrix}   -1 & -3 \\   -5 & -7 \\ \end{bmatrix}

Now, A – B = A + (-B) 

Let’s compute A + (-B).

A + (-B) = \begin{bmatrix}   2 & 4 \\   6 & 8 \\ \end{bmatrix} + \begin{bmatrix}   -1 & -3 \\   -5 & -7 \\ \end{bmatrix} \\ \Rightarrow A + (-B) = \begin{bmatrix}   (2 + (-1)) & (4 + (-3)) \\   (6 + (-5)) & (8 + (-7)) \\ \end{bmatrix} \\ \Rightarrow A + (-B) = \begin{bmatrix}    1 & 1 \\    1 & 1 \\ \end{bmatrix}

Read More,

Solved Problems on Matrix Addition

Problem 1: Perform the addition of the following matrices:

\bold{A = \begin{bmatrix}2 & 3\\ 4& 5\end{bmatrix}}

and \bold{B = \begin{bmatrix}1 & -1\\ -3& 2\end{bmatrix}}

Solution:

To add matrices A and B, we need to add the corresponding elements of each matrix.

A + B = \begin{bmatrix} 2 + 1&3 + (-1)\\ 4 + (-3)& 5 + 2\end{bmatrix}

\Rightarrow A + B =  \begin{bmatrix}3& 2\\1& 7\end{bmatrix}

Therefore, the sum of matrices A and B is \begin{bmatrix}3& 2\\1& 7\end{bmatrix}.

Problem 2: Given the matrices:

\bold{X = \begin{bmatrix} 5 & 2 & -1 \\ 3 & 0 & 4 \end{bmatrix} }        and

\bold{Y = \begin{bmatrix} -2 & 7 & 3 \\ 1 & -1 & 2 \end{bmatrix}}

Calculate the sum of matrices X and Y.

Solution:

To add matrices X and Y, we add the corresponding elements of each matrix.

X + Y = \begin{bmatrix} 5 + (-2) & 2 + 7 & -1 + 3 \\ 3 + 1 & 0 + (-1) & 4 + 2 \end{bmatrix}

\Rightarrow X + Y  = \begin{bmatrix} 3 & 9 & 2 \\ 4 & -1 & 6 \end{bmatrix}

Therefore, the sum of matrices X and Y is  \begin{bmatrix} 3 & 9 & 2 \\ 4 & -1 & 6 \end{bmatrix}.

Problem 3: For matrix P and Q given as follows:

 P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}

Q = \begin{bmatrix} -1 & 0 \\ 2 & -2 \\ -3 & 1 \end{bmatrix}

Compute the sum of matrices P and Q.

Solution:

P + Q = \begin{bmatrix} 1 + (-1) & 2 + 0 \\ 3 + 2 & 4 + (-2) \\ 5 + (-3) & 6 + 1 \end{bmatrix}

\Rightarrow P + Q = \begin{bmatrix} 0 & 2 \\ 5 & 2 \\ 2 & 7 \end{bmatrix}

Therefore, the sum of matrices P and Q is  \begin{bmatrix} 0 & 2 \\ 5 & 2 \\ 2 & 7 \end{bmatrix}.

FAQs on Matrix Addition

Q1: Define Matrix Addition.

Answer: 

Matrix addition is an operation performed on matrices, where corresponding elements of two matrices with same order are added together to form a new matrix.

Q2: How to Add Two or More Matrices?

Answer: 

To add any two or more matrices of same order, we just need to add corresponding element of each matrices with all the other given matrices. For example, the element in the first row and first column of one matrix is added to the element in the first row and first column of the other matrix, and so on. 

Q3: What are the Requirements for Matrix Addition?

Answer: 

Only condition for addition of two matrices is that order of both matrices needs to be same i.e., if one matrix is of order m×n, then other matrix should be of the order m×n. Thus, matrices with different order can’t be added together.

Q4: Can you Add more than Two Matrices together?

Answer: 

Yes, matrix addition can be performed on more than two matrices. To do this, you simply add the corresponding elements of each matrix together. The resulting sum will be the corresponding element of the new matrix.

Q5: What are Properties of Matrix Addition?

Answer:

Some properties of matrix Addition are:

  • Closure Property
  • Commutative Property
  • Associative Property
  • Additive Identity Property
  • Additive Inverse Property

Q6: Is Matrix Addition Commutative?

Answer:

Yes, matrix addition is commutative. This means that changing the order of the matrices being added does not affect the result.



Last Updated : 10 Jun, 2023
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